Affine PBW Bases and MV Polytopes in Rank 2

TL;DR

This paper demonstrates that in rank-2 affine Kac-Moody algebras, MV polytopes defined via different approaches—pre-projective algebras, combinatorics, and PBW bases—are equivalent, providing a unified understanding of these structures.

Contribution

It proves the equivalence of three different definitions of MV polytopes in rank-2 affine cases, unifying various approaches in the theory.

Findings

All three notions of MV polytopes agree in rank-2 affine cases.

Introduces a new characterization of rank-2 affine MV polytopes.

Extends the theory of PBW bases to define MV polytopes in affine cases.

Abstract

Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. Our main tool is a new characterization of rank-2 affine MV polytopes.